Given an elliptic curve of analytic rank at most one, Kolyvagin proved that the Tate-Shafarevich group is finite using his theory of Euler systems. In the higher rank cases, Kolyvagin gave a conjectural description of the structure of the Selmer group in terms of the Euler system. This conjecture has recently been proved by Wei Zhang for a large class of elliptic curves. We shall explain Kolyvagin's conjecture, discuss some of its remarkable consequences and overview the strategy of the proof. To put things in context, we shall begin with a brief survey on the current status of the BSD conjecture.

This is a note prepared for an Alcove Seminar talk at Harvard, Spring 2014. Our main references are [1] and [2].

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Let be an elliptic curve over a number field . Recall that the BSD conjecture relates the arithmetic invariants of with the an analytic object, the -function (see a lowbrow introduction).

Conjecture 1 (BSD)

- has analytic continuation over .
- (Rank conjecture). Denote the
*analytic rank*and the*Mordell-Weil rank*(or,*algebraic rank*) . Then - (BSD formula) Denote . Then Here is the Neron differential of (see the remark), is the local Tamagawa number, is the Tate-Shafarevich group, is a basis of the free part of , is the Neron-Tate height paring on and is the discriminant of .

Remark 1
is called the *period* and is called the *regulator*. Compare the BSD formula with classical class number formula

Remark 2
In general, the module of Neron differentials of is a projective (but not necessarily free) module over : so the global Neron differentials over may not exists. A more elegant way to define the period is then to use adelic integration (with convergence factors). But since we will mainly talk about elliptic curves over (and its base change over an imaginary quadratic field), we will not go into the details.

Remark 3
The BSD conjecture makes sense for general abelian varieties over number fields. The change in the BSD formula is to replace with , with and the height pairing with .

There has been vast progress on the BSD conjecture in recent years. We give a very brief (highly incomplete) summary.

*Analytic continuation*. Due to the recent work of someone in the audience, we now know has analytic continuation for any totally real field of degree .

*Rank*. To make clear the rank part of the conjecture, we recall the -descent exact sequence The -Selmer group is a direct sum of several (denoted by ) copies of and a finite abelian -group. Conjecturally, is finite, and hence conjecturally for any . Consider the following implications for , The implication 1) is trivial. The implication 2) is due to Gross-Zagier and Kolyvagin (80s) for all . The implication 3) is due to Skinner-Urban (2000s) for good ordinary with surjective mod representation . Now consider the analogous statement for rank 1 case, The implication 1) is trivial. The implication 2) is again due to Gross-Zagier and Kolyvagin. The implication 3) is due to Wei Zhang (2013) for good ordinary with surjective under some mild ramification hypothesis.

Remark 4
Notice in both cases, the implications 3) and 2) provide a purely *algebraic* criterion for determining the rank of : together with Bhargava's counting method for 5-Selmer groups, Bhargava-Skinner-Zhang recently proved that at least 66% of all elliptic curves over satisfy the rank part of the BSD conjecture!

*BSD formula*. In the case, the -part of the BSD formula is known for (under similar hypothesis) due to the work of Kato (2000s) and Skinner-Urban on the Iwasawa main conjecture for elliptic curves. In the case, the -part of the BSD formula is proved by Wei Zhang for under similar hypothesis.

Remark 5
Notice nothing general is known for ! Unlike dealing with is not so popular in other fields of number theory, the case is in fact extremely interesting: the major terms contributing the BSD formula indeed come from mysterious powers of 2 and 3. Also, since quadratic twists preserve the 2-torsion structure, understanding the 2-part of the BSD formula is important for understanding the arithmetic of elliptic curves in quadratic twist families (e.g., the congruent number curve family ). In practice, the 2-descent is the easiest to carry out and provides a useful computational tool.

We now brief review the Euler system of Heegner points to motivate the statement of Kolyvagin's conjecture. In the next section, we shall state Wei Zhang's result on Kolyvagin's conjecture and deduce some of its consequences mentioned above.

Let be an elliptic curve over of conductor . Let be an imaginary quadratic extension with discriminant coprime to (we will assume for simplicity). Let be the quadratic character associated to . We assume the *Heegner condition*: every prime factor of splits in . Then and the BSD conjecture predicts that is odd (in particular, ). How do you construct a point?

The key thing is that under the Heegner condition, we have an abundant supply of algebraic points on over the ring class fields of . Recall that classifies cyclic -isogenies between two elliptic curves . For , let be the order of of conductor . Under the Heegner condition, there exists an ideal with norm . Then we have a pair of elliptic curves with CM by , with kernel . This defines a point (a *Heegner point*) , which is defined over the ring class field of by the theory of complex multiplication. Here corresponds to the open compact subgroup of under class field theory (so is the Hilbert class field of ). Using the modular parametrization (mapping to 0) we obtain algebraic points . Taking the trace using the group law on , we construct a Heegner point .

Remark 6
The construction of a Heegner point depends on different choice of ideal classes (we used the trivial class above, but any nontrivial class also works) and the choice of . Nevertheless it is well-defined up to sign and torsion, since the Atkin-Lehner involutions and together act simply-transitively on all Heegner points; is torsion on .

The big theorem of Gross-Zagier we went through last semester is the following.

Theorem 1 (Gross-Zagier)
Here is the Manin constant defined so that , here is the normalized eigenform associated to .

It follows from the Gross-Zagier formula that is infinite order if and only if . When is of infinite order, Kolyvagin used his theory of Euler system to prove the finiteness of . A simple version of his theorem looks like

Theorem 2 (Kolyvagin)
Suppose of infinite order. Let be prime such that is surjective and is not divisible by in (this is true for almost all ). Then and .

The rough idea of Kolyvagin's proof is that he constructed a system of cohomology classes in derived from the Heegner points satisfying nice norm and congruence relations. The local information about these explicit cohomology classes were good enough to bound the Selmer group (via the local and global Tate duality and the Chebotarev density). More precisely, let be the sign of the functional equation, then it was actually shown that and (the latter is contributed by ).

We now briefly recall the construction of Kolyvagin's Euler system. The key thing is the following assumption:

Definition 1
If a prime is inert in , and where is the eigenvalue of the Hecke operator of on the modular from . Then we say is a *Kolyvagin prime*. The positive integer is called the *Kolyvagin index*.

Remark 7
The condition is equivalent to requiring is in the conjugacy class of the complex conjugation in . It follows from the Chebotarev density theorem that there is a positive density set of Kolyvagin primes.

The reason for this key condition is the following. Let Since is inert in , we have .

Then is indeed *invariant* under for . To check this, we need to show that lies in . Namely By assumption , so it remains to show that . This is true since by assumption : the trace of the Heegner point on is exactly the Hecke operator acting on , which projects to . The above relation is also the origin of the name *Euler system*: appears in the Euler factor of at .

More generally,

Definition 3
Let be a square-free product of Kolyvagin primes, then we say is a *Kolyvagin number*. The set of all Kolyvagin numbers is denoted by . We denote . Define where . Then is indeed invariant under the Galois group and *descends* to as long as (because ).

Definition 4
We define the Kolyvagin cohomology class The collection of cohomology classes is called a *Kolyvagin system*.

Notice is nothing but the cohomology class of . In higher rank cases, the class is indeed trivial by Gross-Zagier. Kolyvagin's Euler system constructs more classes in . One can ask whether some of these classes could be nontrivial. An obvious but crucial observation is that this is the same as asking whether some is *not infinitely divisible by *.

Definition 5
For , we define to be the set of all Kolyvagin numbers with exactly prime factors. Define to be the -divisibility of the Kolyvagin cohomology classes , . Namely, is the largest integer such that for all and .

Remark 8
Notice is simply the -divisibility of . In particular, if and only if is of infinite order.

Remark 9
One can describe the action of complex conjugation on these Kolyvagin classes depending on , the number of prime factors of : lies in the -eigenspace (e.g., when , lies in the -eigenspace).

Kolyvagin proved that in fact So increasing the number of prime factors of may help bring down the -divisibility! We define

The hope is that even in the case that is torsion, the cohomology classes eventually becomes nontrivial.

Assuming this key conjecture, Kolyvagin's work allows us to understand the refined structure of .

Starting from a nontrivial element by assumption , Kolyvagin constructed the whole -Selmer group (even in the higher rank case!). Moreover, its -eigenspace can be completely determined.

Theorem 3 (Kolyvagin)
Suppose . Let (so ).

- is contained in the subgroup of generated by .
- ; and has the same parity as . Let be the deficiency (so is even).
- Let be the finite part of . Write where are non-increasing. Then and In particular, , with equality if and only if .

Now we are ready to state Wei Zhang's theorem precisely and then deduce some remarkable consequences.

Theorem 4 (Wei Zhang)
Suppose is good ordinary. Suppose is surjective and ramifies at any . Then . In particular, Kolyvagin's conjecture is true.

Remark 12
There is a similar version of this theorem for Euler systems of Heegner points on Shimura curves. Both versions are vital in the argument.

Corollary 2
Under the assumption of Theorem 4. If , then and is finite.

Proof
One can choose an imaginary quadratic field satisfying the Heegner hypothesis and for the quadratic twist . Then by Gross-Zagier and Kolyvagin, and is finite. So . Therefore . Now by Kolyvagin's Theorem 3, b), is equal to rank of the larger piece of , which implies . So the desired result follows from Corollary 1.
¡õ

Remark 13
Using the general version of Theorem 4 for Shimura curves, one can relax the ramification assumption for this corollary: e.g., when is square-free, one only needs to assume that is ramified at any such that .

Corollary 3
Under the assumption of Theorem 4. If . Then the -part of the BSD formula is true for .

Proof
Since comparing with the Gross-Zagier formula we know that the BSD formula for is equivalent to It is known in general that that the any prime factor of Manin constant must divide , so is coprime to . Under the assumption ramifies at , each is coprime to . So the -part of the BSD formula is equivalent to up to -adic units. By Theorem 3 c), this is equivalent to Namely, . This is exactly Theorem 4.
¡õ

Remark 14
From the above argument we see that the BSD formula in the rank 1 can be translated into a (naive-looking) question about the divisibility of (derived) Heegner points, thanks to the Gross-Zagier formula. We also see that the apparently stronger Theorem 4 is indeed not stronger: is precisely predicted by the BSD conjecture.

Wei Zhang's proof beautifully blends various ideas and ingredients:

- "Deforming" the -Selmer group via congruences. For suitable level raising primes , Ribet and Diamond-Taylor's level raising theorems allow one to find a newform of level that is congruent to the original newform of level mod . In other words, one can keep the same and vary the -Selmer groups while only changing one local condition at at a time.
- When the level raising prime is chosen to be inert in , the root number changes. Gross-Parson proves that the -Selmer parity changes as predicted (where the ramification assumption and is needed). Together with the Chebotarev density argument, one is able to lower the -Selmer rank via lever-raising. In the rank one case, Gross-Parson relates the level raising -Selmer rank with the local -divisibility of the generator of at .
- To actually show that is the generator of , one uses the Jochnowitz congruence, which relates the local -divisibility of the Heegner point at with the so-called algebraic part mod . To establish the Jochnowitz congruence, one needs the multiplicity one of the Hecke module of supersingular points on localized at a non-Eisenstein ideal, provided by Mazur's principle; and Gross's explicit Waldsburger formula on the definite quaternion algebra ramified at .
- To compute , one uses Skinner-Urban's BSD formula in the rank 0 case (where the good ordinary assumption is needed at the moment, but may be removed later). Showing is nonzero mod needs the comparison between the congruence ideal of , the modular degree of and the component group of the Neron model of at . This was established by Ribet and Takahashi.
- One then reduces the higher rank cases to the rank 1 case by successively lever-raising and rank lowering. To recover the nonvanishing of Kolyvagin's system from the rank one case, one uses a "triangularization" of -Selmer group and the following cohomological congruences between Heegner points: the localization at of on is congruent to the localization at of on the Shimura curve . This is a combination of Jochnowitz congruences and Bertolini-Darmon congruences (used in the their proof of the anti-cyclotomic Iwasawa main conjecture). The latter asserts that the localization at of Heegner points on indeed factors through the component group at , hence can be linked to the reduction of Heegner points on the components of the Cerednik-Drinfeld fiber , which matches up with the Hecke module of supersingular points on the definite quaternion algebra ramified at . The cohomological congruence is then established by the multiplicity one.

[1]Selmer groups and the divisibility of Heegner points, 2013, www.math.columbia.edu/~wzhang/math/online/Kconj.pdf.

[2]Kolyvagin's work on modular elliptic curves, $L$-functions and arithmetic (Durham, 1989), London Math. Soc. Lecture Note Ser., 153 Cambridge Univ. Press, Cambridge, 1991, 235--256.